Let $H$ be a graph with at least one edge. Define ex $(n ; H)$, where $n$ is an integer with $n \geqslant|H|$. Without assuming the Erdős-Stone theorem, show that the sequence $\operatorname{ex}(n ; H) /\left(\begin{array}{l}n \\ 2\end{array}\right)$ converges as $n \rightarrow \infty$.

State precisely the Erdős-Stone theorem. Hence determine, with justification, $\lim _{n \rightarrow \infty} \operatorname{ex}(n ; H) /\left(\begin{array}{c}n \\ 2\end{array}\right) .$

Let $K$ be another graph with at least one edge. For each integer $n$ such that $n \geqslant \max \{|H|,|K|\}$, let

$$f(n)=\max \{e(G):|G|=n ; H \not \subset G \text { and } K \not \subset G\}$$

and let

$$g(n)=\max \{e(G):|G|=n ; H \not \subset G \text { or } K \not \subset G\} .$$

Find, with justification, $\lim _{n \rightarrow \infty} f(n) /\left(\begin{array}{l}n \\ 2\end{array}\right)$ and $\lim _{n \rightarrow \infty} g(n) /\left(\begin{array}{l}n \\ 2\end{array}\right)$.